## 1. Introduction

[2] The Soil Conservation Service (currently the Natural Resources Conservation Service) curve number (SCS-CN) [*U.S. Soil Conservation Service* (*SCS*), 1972] method is widely used to predict runoff quantities in ungauged basins. While more sophisticated methods are available, its simplicity and dependence on readily available catchment properties has contributed to its continued popularity, particularly among practicing water resource engineers [*Ponce and Hawkins*, 1996; *Garen and Moore*, 2005]. The basic form of the SCS-CN rainfall-runoff relationship appears to be logical in that no runoff occurs until a threshold in rainfall is met and then, the fraction of rainfall contributing to runoff increases as the rainfall amount becomes larger:

where *Q* is event discharge volume (mm), *P* is effective rainfall (mm), and *S* is a conceptual available soil storage volume (mm) calculated from

where *Cn* varies between 0 (no runoff generation) and 100 (all rain produces runoff) and is traditionally based on catchment land use and soil type via published tables [e.g., *SCS*, 1972]. Most hydrologic engineering texts and manuals still suggest that the *Cn* can be adjusted to account for wetter or drier conditions as determined by the 5-day antecedent rainfall although the Natural Resources Conservation Service has discounted the accuracy of using antecedent rainfall to adjust the *Cn*.

[3] For this standard formulation of the SCS-CN method, the probability of a *Q* of a given magnitude, *Q*_{i}*,* is related to the probability of the causative *P* of a given magnitude *P*_{i}:

where *g* indicates a function relating *Q* and *P*, in this case *g* = equation (1).

[4] However, a limitation of this standard formulation stems from the hydrological reality that *S* itself should not be an immutable parameter but one that should vary with changes in catchment soil moisture storage, a fact only grossly captured by the antecedent rainfall adjustment. Recent work has focused on refining the SCS-CN method for more conceptually coherent use in continuous watershed models by incorporating an underlying soil moisture accounting scheme [i.e., *Michel et al.*, 2005]. But, in many engineering design applications of the SCS-CN method, say, sizing a culvert, engineers are interested in estimating a given discharge associated with a desired return period. If the variability of the *S* due to antecedent wetness conditions is ignored when estimating event-based probabilities of runoff, the SCS-CN method implies that the probability of a given runoff quantity is dictated only by the probability of the causative precipitation input, as indicated in (3), an assumption countered by studies of flood processes [*Merz et al.*, 2006]. Additionally, precipitation-discharge (*P*-*Q*) plots many times reveal no consistent 1:1 relationship between *P* and *Q*, e.g., a single precipitation amount can have multiple resulting discharges. Sizable scatter in *P*-*Q* pairs is frequently noted in the literature, many times leading to rank ordering and recombining of *P* and *Q* observations so each value in the pair has the same return period instead of preserving the natural *P*-*Q* pairing [e.g., *Hawkins*, 1993]. As an example of a *P*-*Q* plot with scatter, Figure 1 shows data from the Fall Creek watershed near Ithaca, New York (additional details on the selection of these *P*-*Q* pairs is given below). To give a sense of the variation in *Cn* that capture these data, we best fit (1) to the data (*Cn* = 82) and calculated *Cn* that envelope 80% of the points, corresponding to *Cn* = 71 (bottom line in Figure 1) and *Cn* = 90 (top line in Figure 1).

[5] Thus, while hydrologic engineers have generally recognized that the antecedent moisture state influences storm runoff quantity, nobody has satisfactorily proposed a means to include the moisture state in the determination of runoff risk. *McCuen* [2002] suggested treating the *Cn* (or *S*) as a random variable, identifying a suitable distribution and deriving confidence intervals. *De Michele and Salvadori* [2002] incorporated soil moisture into the calculation of the frequency distribution of peak floods but only calculated the probability of being in one of the three published antecedent moisture conditions (AMC) as based on antecedent rainfall. Conversely, *Mishra and Singh* [2006] developed a relationship between *S* and 5-day antecedent rainfall for watersheds in India but did not try to relate variations in *S* to changes in risk of a given *Q*. Building on elements of this previous work, we contend that *S* should vary with a watershed's antecedent wetness [e.g., *McCuen*, 2002], that the distribution of *S* can be deterministically developed [e.g., *Mishra and Singh*, 2006], and that *S* can be incorporated into estimates of risk of runoff, as done in a limited way by *De Michele and Salvadori* [2002]. However, differing from this previous work, we suggest that *S* is best linked to base flow immediately preceding the causative precipitation event, *Q*_{base}, instead of the 5-day antecedent rainfall, at least in temperate regions with large intra-annual variations in evapotranspiration. For a catchment in Belgium, *Troch et al.* [1993] related streamflow to catchment water table height using Boussinesq groundwater theory, thereby indicating the possibility of using base flow as a proxy for *S* within the framework of the SCS-CN method.

[6] The central aim of this paper is to present a simple approach for modifying the SCS-CN method to make more robust estimates of runoff risk. This modified method treats runoff generation (*Q*) as the outcome of a bivariate process involving interaction between watershed storage, *S*, which is a function of antecedent moisture conditions in the watershed, and precipitation, *P*. Treating the magnitude of *Q* as the outcome of a bivariate process:

where

and *f*_{P,S} is a bivariate probability density function and *h* is a function relating both *P* and *S* to *Q,*(1) in this case. Fundamentally, (4) calculates the volume (probability) under regions of the joint probability distribution where the given pairing of *P* and *S* result in a desired *Q*.

[7] Quantification of hydrologic events using bivariate formulations has started to gain favor in the last decade but has largely been limited to relating flood peak discharge and event volume [see *Yue and Rasmussen*, 2002]. Here, we handle the interaction between more fundamental watershed processes using such a bivariate formulation.